A NEW VARIATIONAL FRAMEWORK FOR LARGE STRAIN …

Keynote Lecturer: Javier Bonet A new variational framework for large strain hyperelastic piezoelec-tric materials

V International Conference on Computational Methods for Coupled Problems in Science and EngineeringCOUPLED PROBLEMS 2013

S. Idelsohn, M. Papadrakakis and B. Schrefler (Eds)

A NEW VARIATIONAL FRAMEWORK FOR LARGESTRAIN PIEZOELECTRIC HYPERELASTIC MATERIALS

J. BONET∗, R. ORTIGOSA∗ AND A. J. GIL∗

∗Civil & Computational Engineering Centre, College of Engineering, Swansea University,Swansea SA2 8PP, UK

Key words: Energy harvesting; Piezoelectricity; Polyconvexity; Large deformations;Finite Element Method

Abstract. In this paper, a novel nonlinear variational formulation is presented for thenumerical modelling of piezo-hyperelastic materials. Following energy principles, a newfamily of anisotropic extended internal energy density functionals is introduced, depen-dent upon the deformation gradient tensor and the Lagrangian electric displacement fieldvector. The requirement to obtain solutions to well defined boundary value problems leadsto the definition of energy density functionals borrowing concepts from polyconvex elas-ticity. Material characterisation of the constitutive models is then carried out by means ofexperimental matching in the linearised regime (i.e. small strains and small electric field).The resulting variational formulation is discretised in space with the help of the FiniteElement Method, where the resulting system of nonlinear algebraic equations is solved viathe Newton-Raphson method after consistent linearisation. Finally, a series of numericalexamples are presented in order to assess the capabilities of the new formulation.

1 INTRODUCTION

The earliest piezoelectric materials to be discovered, i.e. crystals, have shown limitedapplicability due to their high stiffness and brittleness. The recent advent of piezoelec-tric polymers has meant a turning point in the development of piezoelectricity. Thecircumvention of the drawbacks associated with their crystal predecessors has broadenedconsiderably their applications as actuators, power generators and energy harvesters.

Piezoelectric polymers have traditionally been used as smart actuators in microelec-tromechanical systems. However, their ability to emulate the functioning of biologicalmuscles as well as their large strain capabilities, have recently triggered the emergenceof new exciting applications, such as artificial muscles. A more recent application withinthe field of smart actuators can be found in space microwave antennas. These devices ex-perience shape deviations due to pre-stress and thermal expansion. These deviations are

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J. Bonet, R. Ortigosa and A. J. Gil

corrected by controlling an applied electric potential on a piezoelectric patch to maintainthe desired shape of the antenna reflector. Figure 1 illustrates two application examples.

Figure 1: (a) Electroactive actuated robotic arm. (b) Shape control of space antenna.

The large strain and piezoelectric capabilities associated to piezoelectric polymers con-fer them with attractive properties in the field of power generation and energy harvesting.There is currently a growing need for this kind of applications. For instance, piezoelectriceels (see Figure 2(a)), which are used as part of submarine devices in long endurance mili-tary missions. The energy-harvesting eel is designed to extract energy from the wake of abluff body in an ocean current. The basic configuration is a leading bluff body trailed bya thin flexible piezoelectric eel. The bluff body generates vortices which excite a flappingmotion of the eel. The eel deformation results in strain of the piezoelectric membrane,which in turn generates a voltage across the material. Other current applications includesmall piezoelectric devices that when attached to the insole of a shoe are able to transformmechanical motion into electrical power (see Figure 2(b)).

Figure 2: (a) Piezoelectric eel. (b) Walking energy harvesting device.

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Electromechanical interactions in materials are not only due to piezoelectricity. An-other important phenomenon which has found a vast number of applications within thefield of electromechanical actuators is electrostriction. This phenomenon results from aquadratic relation between stresses and electric field. The application of electric fieldson purely electrostrictive materials would lead to a deformation on the material. How-ever, unlike piezoelectricity, this effect cannot be reversed. It is worth emphasising thatpiezoelectricity and electrostriction are not mutually exclusive.

The existing framework for the numerical simulation of piezoelectric materials requiresan enhancement as a result of the development of these new polymers, capable of under-going large deformations. Unlike crystals, the classical linearised theory can no longer beapplied for a reliable computer simulation. In this paper, a nonlinear variational formu-lation for piezo-hyperelastic materials is introduced with the help of the internal energydensity U constructed on the basis of the right Cauchy-Green deformation tensor C andthe Lagrangian electric displacement field vector D0.

2 ANISOTROPIC STRUCTURE OF PIEZOELECTRIC MATERIALS.

The proposed internal energy density U must lead to the definition of a constitutivemodel which needs to capture the material response in an accurate manner. Accordingly,U must be compliant with the physical and mathematical constraints inherent to thesematerials (e.g. anisotropy). This will be taken into consideration by means of a hybridapproach which combines the isotropic extension surface concept introduced in reference[2] and a more recent methodology proposed in reference [3].

The first approach, i.e. the isotropic extension concept, introduces the so-called isotropicextension surface φ, which includes the set of vectors and tensor valued functions whichare preserved under the action of the material symmetry group which characterises thecorresponding anisotropy of the material, that is,

φ = φ (C,D0) . (1)

As a result, the creation of anisotropic invariants can alternatively be achieved by for-mulating isotropic invariants which take into consideration the isotropic extension surfaceφ. Thus, the extended internal energy density functional U is created in terms of isotropicinvariants of the following arguments,

U = U (C,D0, φ (C,D0)) . (2)

An alternative approach for the creation of anisotropic invariants was recently proposedin [3]. In this reference, a new metric tensorG = HHT is introduced, whereH representsa linear tangent map established between the Cartesian base triad e1, e2, e3 and thecrystallographic base system a1,a2,a3, the latter related to the associated Bravaislattice (see Figure 3).

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Figure 3: Crystallographic a1,a2,a3 and Cartesian e1, e2, e3 base vectors.

3 CONSTITUTIVE EQUATIONS IN ELECTRO-MECHANICS.

The derivation of the constitutive equations for Electro-Mechanics requires a revision ofthe First and Second laws of Thermodynamics. Following an integral material description,the First law of Thermodynamics gives the variation of internal energy as [1],

d

dt

∫

Ω0

U (F ,D0, η0) dΩ0 =

∫

Ω0

P : F dΩ0 +

∫

Ω0

E0 · D0 dΩ0

−∫

∂Ω0

∇0 · q0 dΩ0 +

∫

Ω0

R0 dΩ0, (3)

where F is the deformation gradient tensor, D0 is the Lagrangian electric displacementfield, η0 is the Lagrangian entropy, P is the first Piola-Kirchoff stress tensor, E0 is theLagrangian electric field, θ is the temperature, q0 is the Lagrangian Fourier-Stokes heatflux vector field and R0 is the Lagrangian heat supply field per unit volume.

The Clausius-Duhem form of the Second law of Thermo-Electro-Mechanics can bewritten as [1],

d

dt

∫

Ω0

η0 dΩ0 ≥∫

Ω0

R0

θdΩ0 −

∫

Ω0

∇0 ·q0

θdΩ0. (4)

Combination of both First and Second laws leads to,

∫

Ω0

(P − ∂U

∂F

): F dΩ0 +

∫

Ω0

(E0 −

∂U

∂D0

)· D0 dΩ0

+

∫

Ω0

(θ − ∂U

∂η0

)η0 dΩ0 −

∫

Ω0

1

θ∇0θ · q0 dΩ0 ≥ 0. (5)

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Constitutive equations based on the internal energy U can be derived from the combi-nation of the First and Second law of Thermo-Electro-Dynamics [1],

P =∂U (F ,D0, η0)

∂FE0 =

∂U (F ,D0, η0)

∂D0

θ =∂U (F ,D0, η0)

∂η0− 1

θ∇0θ · q0 ≥ 0, (6)

where q0 can be related to the material gradient of the temperature θ as,

q0 = −k∇0θ, (7)

where k is the thermal conductivity tensor. Positive definiteness of k ensures that−1

θ∇0θ · q0 ≥ 0. For reversible Electro-Mechanics, the constitutive equations can be

simplified as,

P =∂U (F ,D0)

∂FE0 =

∂U (F ,D0)

∂D0

. (8)

Different constitutive laws could be proposed by introducing alternative internal energydensities U . In addition, the electric enthalpy H (F ,E0) can be introduced by means ofthe Legendre transform,

H (F ,E0) = U (F ,D0)−E0 ·D0. (9)

For the electric enthalpy, analogous constitutive laws to equations (8) can be writtenas,

P =∂H (F ,E0)

∂FD0 = −∂H (F ,E0)

∂E0

. (10)

In the abscence of external electric or mechanical loads, equilibrium configurationsare reached when the internal energy U (F ,D0) attains a minimum. It is possible toshow that for the same equilibrium configuration, the complementary electric enthalpyH (F ,E0) attains a saddle point. In other words, in a neighbourhood of the equilibriumconfiguration, the internal energy U (F ,D0) would be a convex function of F and D0

whereas the electric enthalpy H (F ,E0) would be a saddle function, i.e convex in Fand concave in D0. Figure 3 shows the nature of both U (F ,D0) and H (F ,E0) in aneighbourhood of the equilibrium configuration for a one-dimensional case.

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(a) Internal energy (b) Electric enthalpy

Figure 4: Behaviour of different types of energy densities in the vicinity of an equilibrium configuration.

4 Mathematical requirements: Polyconvexity.

The properties of the constitutive models used for the numerical simulation of piezo-electric polymers must also guarantee existence of solutions to boundary value problemsin large deformation regimes. Convexity of the internal energy functional ensures exis-tence. However, this condition is too stringent as it precludes the appearance of differentequilibrium configurations with equal potential energy, i.e. buckling. Moreover, it canalso violate material frame indifference. A necessary and almost sufficient condition forthe existence of minimisers to a variational formulation is quasiconvexity. However, this isan integral condition whose verification is a cumbersome task. As a result, the conditionspreferred in this work which guarantee the existence of solutions to well defined boundaryvalue problems, are polyconvexity and coercivity [4]. Borrowing concepts from nonlin-ear elasticity, an energy density functional is said to be polyconvex in the deformationgradient tensor F if it can be written as a convex function of the following arguments,

U = U (F ,CofF , detF ) . (11)

An intuitive and physical understanding of polyconvexity can be gained from its ge-ometric interpretation. The arguments appearing in equation (11) govern the transfor-mations by means of which line, area and volume material elements are mapped to theirspatial counterparts (see Figure 5).

There is an extensive work done in the creation of polyconvex energy functionals in thefield of nonlinear elasticity. However, the coupled nature of the electromechanical problemrequires a generalisation of this concept. As a result, this paper presents a methodologyfor the development/implementation of polyconvex energy functionals on the basis ofthe deformation gradient tensor F and the material electric displacement field D0 [5].According to this methodology, polyconvexity is enforced by means of a convex function

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Φ

Figure 5: Deformation mapping and relevant kinematic operators (F ,CofF , J = detF ).

of the following arguments,

U = U (F ,CofF , detF ,D0) . (12)

5 Constitutive models for piezo-hyperelastic materials.

From the physical point of view, a possible decomposition of the internal energy densityis carried out in terms of the three constitutive parts involved in the formulation, i.e.mechanical, piezoelectric and electrostrictive,

U (D0,C) = Um (C) + Ue (D0,C) + Up (D0,C) , (13)

where Um, Up and Ue account for the mechanical, piezoelectric and electrostrictive com-ponents of the stored internal energy functional U , respectively. Naturally, the mechanicalcomponent of the energy functional can also be split into its isotropic and anisotropic con-tributions,

Um = Uiso,m (C) + Uaniso,m (C) . (14)

The isotropic contribution of the mechanical component Uiso,m (C) can be chosen fromany of the existing polyconvex strain energy functionals, i.e. Mooney-Rivlin. The defi-nition of the three remaining components (see equations (13)-(14)) completes the char-acterisation of the material model. Anisotropic mechanical energy functionals built onthe basis of polyconvex invariants can be found in [3]. This author proposes the follow-

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ing expression for Uaniso,m in terms of an additive decomposition of a series of families jcharacterising the anisotropy of the material,

Uaniso,m =∑j

µj

(J4j

αj+1

αj + 1+

Jβj+15j

βj + 1+

tr(Gm

j

)III

−γjC

γj

), (15)

with,

J4j = tr(CGm

j

); J5j = tr

(C CofGm

j

); IIIC = detC, (16)

where Gmj is a possible metric tensor characterising the anisotropy of the material,

αj, βj and γj are dimensionless parameters and µj is a mechanical material parameter.For the material model to be complete, it is necessary to account for the remainingconstitutive counterparts, i.e. piezoelectric and electrostrictive.

In this paper, a novel definition for the respective components of the energy functionalon the basis of polyconvex invariants is introduced. A possible electrostrictive componentUe would be,

Ue =∑j

ωj

[pe,j ·Cpe,j + I2C +

(pe,j · pe,j

)2]+ fe (C) + ge (D0) , (17)

with,

pe,j = mjGejD0 D0 =

D0√µrefε0

, (18)

where Gej is a possible metric tensor characterising the anisotropy of the material, mj

is a dimensionless parameter, ωj is an electrotrictive material parameter, ε0 is the electricpermittivity in vacuum and µref is a reference mechanical material parameter. A possiblecontribution for the piezoelectric component Up would be,

Up =∑j,k

λjk

[pp,jk ·Cpp,jk + I2C +

(pp,jk · pp,jk

)2]+ fp (C) + gp (D0) , (19)

with,

pp,jk = sjGpjD0 +N k, (20)

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where Gpj is a possible metric tensor characterising the anisotropy of the material,

sj is a dimensionless parameter, N k accounts for the structural vectors included in theisotropic extension surface of the material considered and λjk is a piezoelectric materialparameter. It is important to emphasise that the definitions of the mechanical Um andelectrostrictive Ue components are based upon the methodology proposed in [3]. How-ever, the piezoelectric component has been formulated by means of a combination of thismethod and the isotropic extension concept in [2].

The functions fe, fp, ge and gp are chosen so that no stress or electric field is obtainedin the origin,

S|D0=0,C=I = 2∂Up

∂C

∣∣∣∣D0=0,C=I

+ 2∂Ue

∂C

∣∣∣∣D0=0,C=I

= 0

E0|D0=0,C=I =∂Up

∂D0

∣∣∣∣D0=0,C=I

+∂Ue

∂D0

∣∣∣∣D0=0,C=I

= 0. (21)

The complete characterisation of the material model defined through the internal en-ergy density requires the definition of the material properties αj, βj, γj, µj, ωj, mj, λjk,sj and the metric tensors Gm

j , Gej and Gp

j . This is achieved by performing a match in theorigin, i.e. C = I,E0 = 0, between the constitutive tensors derived from the proposedenergy functionals and the experimental tensors available in the linearised regime (i.e.small strains and small electric field),

C|C=I,E0=0 = c; P |C=I,E0=0 = p; A|C=I,E0=0 = ε, (22)

where C, P and A are the elastic, piezoelectric, and dielectric material tensors, respec-tively, derived from the energy functional as follows,

Aij (C,E0) =

[∂2U (C,D0)

∂D0∂D0

]−1

ij

Pijk (C,E0) = −2Ami∂2U (C,D0)

∂Cjk∂D0m

Cijkl (C,E0) = 4∂2U (C,D0)

∂Cij∂Ckl

+ 2∂2U (C,D0)

∂Cij∂D0m

Pmkl. (23)

From the previous match, the necessary material parameters can be obtained either byidentification or minimisation.

6 Numerical results.

The resulting variational formulation is discretised in space with the help of the FiniteElement Method, where the resulting system of nonlinear algebraic equations is solved

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via the Newton-Raphson method after consistent linearisation. In this section, a series ofnumerical examples are presented in order to demonstrate the robustness and applicabilityof the formulation.

Figure 6 shows an example in which the application of an external electric field pro-duces a deformation in a composite shell. Figure 7 depicts another example of actuatorapplication. The application of an external electric field on a composite material producesdeformations and changes in shape. In this case, the different anisotropic orientation ofboth layers of materials leads to an out of plane deformation. This effect enables thesematerials to be designed to carry out a specific mechanically demanded task.

REFERENCES

[1] Gonzalez, O. and Stuart, A.M. A first course in continuum mechanics. Cambridge(2008).

[2] Xiao, H. On isotropic extension of anisotropic tensor functions. Z. angew. Math.Mech. (1996) 76:205–214.

[3] Schroder, J. Neff, P. and Ebbing, V. E. Anisotropic polyconvex energies on the basis ofcrystallographic motivated structural tensors. Journal of the Mechanics and Physicsof solids (2008) 56:3486-3506.

[4] Ball, J.M. Convexity conditions and existence theorems in nonlinear elasticity. Arch.Rat. Mech. Anal. (1977) 63:337–403.

[5] Rogers, R.C. Nonlocal variational problems in nonlinear electromagneto-elastostatics.SIAM J. Math. Anal. (1988) 19:1329-1347.

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(a) Electric potential (V )

(b) Electric field in OZ direction (V/m)

(c) Stress σxx (N/m2)

Figure 6: Composite circular shell subject to electric potential gradient across the thickness.

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(a) Electric potential (V )

(b) Electric displacement in OX direction (N/mV )

(c) Displacement in OZ direction (m)

Figure 7: Composite rectangular shell subject to electric potential gradient across the thickness.

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A NEW VARIATIONAL FRAMEWORK FOR LARGE STRAIN …